Dynamical Origin of Spectral Rigidity and the Montgomery–Dyson Correspondence: Part III. Semiclassical Orbit Selection Hypothesis and Wave-Dynamical Interference in the Gutzwiller Trace Formula

While the Montgomery–Dyson correspondence revealed a profound statistical equivalence between the non-trivial zeros of the Riemann zeta function and the eigenvalue spectra of complex quantum systems, its microscopic dynamical origin remains an open question. In semiclassical physics, the Gutzwiller trace formula establishes a semiclassical correspondence between quantum spectral density and a sum over classical periodic orbits, paralleling the explicit formulas of prime number theory (Riemann–von Mangoldt–Weil) that express prime distributions through a sum over zeta zeros. However, both formalisms treat the underlying orbits kinematically, evaluating the sum over a pre-existing classical manifold without addressing how wave-dynamical interference across adjacent orbits influences long-term spectral rigidity. Following the foundational wave-node incommensurability framework of Part I and the empirical 238U intervention analysis of Part II, this paper (Part III) proposes a semiclassical orbit-selection hypothesis based on wave-node interference. Within this proposed framework, commensurate periodic orbits characterized by rational period ratios (T_A / T_B ∈ Q) permit stationary phase alignment (δ_min ≤ δ_tol), facilitating multi-channel phase locking that can induce destructive parametric dephasing across adjacent semiclassical contributions. Conversely, incommensurate orbits exhibit persistent wave-node slipping (δ_min > δ_tol), suppressing secular phase accumulation and providing an intrinsic resonance-avoidance barrier. Consequently, we propose that mutually incommensurate, prime-like periodic orbits can preferentially contribute to sustaining coherent spectral rigidity. By connecting arithmetic non-divisibility with continuous resonance avoidance, this work presents a wave-dynamical perspective linking periodic-orbit expansions and the Montgomery–Dyson correspondence.